Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

تمرين 21

This exercise shows that the OLS estimator of a subset of the regression coefficients is consistent under the conditional mean independence assumption stated in Appendix 7.2. Consider the multiple regression model in matrix form Y=Xß + Wy + u, where X and W are, respectively, n × k 1 and n × k 2 matrices of regressors. Let X i and W i denote the i th rows of X and W [as in Equation (18.3)]. Assume that (i)
, where is a k 2 × 1 vector of unknown parameters; (ii) (Xi, W i Yi) are i.i.d.; (iii) (X i W i u i ) have four finite, nonzero moments; and (iv) there is no perfect multicollinearity. These are Assumptions #l-#4 of Key Concept 18.1, with the conditional mean independence assumption (i) replacing the usual conditional mean zero assumption.
a. Use the expression for
given in Exercise 18.6 to write
- ß =
.
b. Show that
where
=
, and so forth. [The matrix
if
: for all i,j, where A n,ij and A ij are the (i, j) elements of A n and A.]
c. Show that assumptions (i) and (ii) imply that
.
d. Use (c) and the law of iterated expectations to show that

e. Use (a) through (d) to conclude that, under conditions (i) through
(iv)